Alberto Gandolfi

A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy.

In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study is mainly focused on the effect of a therapy that induces tumour vessel loss (anti-angiogenic therapy), with the aim of finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, if tumour and/or vessel dynamics exhibit time delays, we derive conditions for the existence of Hopf bifurcations. The destabilizing effect of delays on achieving the tumour eradication is also investigated. Finally, global conditions for stability and eradication in the presence of delays are given for some particular cases.

A free boundary problem with unilateral constraints describing the evolution of a tumor cord under the influence of cell killing agents

A system of tumor cords is schematized by an array of identical cords, each one having approximately a rotational symmetry around its central blood vessel. A mathematical model for the evolution of the cord is presented, taking into account the influence of a limiting nutrient on the proliferation and death of the cells, the volume reduction of the necrotic material due to fluid loss from the cord, and the influence of chemotherapy or radiation treatment. Both the steady state and the evolution problem are considered, showing existence and uniqueness of the solution. A peculiar feature of the evolution model is that the boundary conditions for nutrient concentration on the interface between viable cord and the necrotic region may change during the response to treatment, depending on whether or not new cells enter the necrotic region.

Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit?

Although commonly related to nutrient deprivation, the cause of the formation of the necrotic core in the multicellular tumour spheroids is still a controversial issue. We propose a simple model for the cell ATP production that assumes glucose and lactate as the only fuel substrates, and describes the main reactions occurring in the glycolytic and the oxidative pathways. Under the key assumption that cell death occurs when ATP production falls to a critical level, we formulate a multiscale model that integrates the energy metabolism at the cellular level with the diffusive transport of the metabolites in the spheroid mass. The model has been tested by predicting the measurements of the necrotic radius obtained by Freyer and Sutherland (1986a) in EMT6/Ro spheroids under different concentrations of glucose and oxygen in the culture medium. The results appear to be in agreement with the hypothesis that necrosis is caused by ATP deficit.

Mathematical models of the cell cycle with a view to tumor studies

This paper gives a review of mathematical models of cell populations. Attention is focused on the models containing a description of the cycle or even of the biochemical events controlling cell growth. An effort is made to present the material in a consistent framework, and to bring into evidence possible connections. The applications of the models to the analysis of experimental data from tumor cell populations are particularly stressed.

Cell kinetics in tumour cords studied by a model with variable cell cycle length.

A mathematical model is developed that describes the proliferative behaviour at the stationary state of the cell population within a tumour cord, i.e. in a cylindrical arrangement of tumour cells growing around a blood vessel and surrounded by necrosis. The model, that represents the tumour cord as a continuum, accounts for the migration of cells from the inner to the outer zone of the cord and describes the cell cycle by a sequence of maturity compartments plus a possible quiescent compartment. Cell-to-cell variability of cycle phase transit times and changes in the cell kinetic parameters within the cord, related to changes of the microenvironment, can be represented in the model. The theoretical predictions are compared against literature data of the time course of the labelling index and of the fraction of labelled mitoses in an experimental tumour after pulse labelling with 3H-thymidine. It is shown that the presence of cell migration within the cord can lead to a marked underestimation of the actual changes along cord radius of the kinetics of cell cycle progression.

The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation

In the first part of this paper we study the properties of a class of linear differential equations with periodic real coefficients and show that the structure of this family of systems guarantees the unstability of the null solution. In the second part, we use this property to demonstrate that antiangiogenic drugs that act only inhibiting the endothelial cell proliferation might be uneffective in tumor elimination under some relevant biological conditions. Finally, we derive some sufficient criteria that guarantee the disease elimination by therapies using this class of drugs.

Pharmacokinetic analysis of azelaic acid disodium salt. A proposed substrate for total parenteral nutrition

SummaryAzelaic acid was the first dicarboxylic acid proposed as an alternative energy substrate in total parenteral nutrition. In this study, the pharmacokinetics of azelaic acid were investigated in 12 healthy volunteers, 7 receiving a constant infusion (10g over 90 min) and 5 a bolus dose (1g). The 24h urinary excretion and plasma concentration in blood samples taken at regular intervals were assayed by gas-liquid chromatography. Experimental data were analysed by a 2-compartment nonlinear model that describes both tubular secretion and cellular uptake in Michaelis-Menten terms. A high value of urinary excretion (mean 76.9% of infused dose) and a mean clearance of 8.42 L/h were found, suggesting the presence of tubular secretion. Estimating the population mean of the pharmacokinetic model parameters gave a maximal cellular uptake of 0.657 g/h. The model predicts that 90% of the maximal uptake should be reached in the plateau phase of a constant infusion of 2.2 g/h. The presence of extensive and rapid losses through urinary excretion, and the low estimated value of the maximal cellular uptake, indicate that azelaic acid is not suitable as an energy substrate for total parenteral nutrition.

An age-structured model of epidermis growth

We propose a model with age and space structure for the evolution of the supra-basal epidermis. The model includes different types of cells: proliferating cells, differentiated cells, corneous cells, and apoptotic cells. We assume that all cells move with the same velocity and that the local volume fraction, occupied by the cells is constant in space and time. This hypothesis, based on experimental evidence, allows us to determine a constitutive equation for the cell velocity. We focus on the stationary case of the problem, that takes the form of a quasi-linear evolution problem of first order, and we investigate conditions under which there is a solution.

Use of bispecific hybrid antibodies for the development of a homogeneous enzyme immunoassay

Hybrid bispecific monoclonal antibodies reacting with carcinoembryonal antigen (CEA) and with the E. coli enzyme beta-galactosidase (GZ) were produced by fusion of hybridomas or chemical linkage of half-antibodies. Since the original anti-GZ antibody used in these experiments was capable of protecting GZ from thermal denaturation, it was possible, by hybridizing it with two different non-competitive anti-CEA antibodies, to design a homogeneous enzyme immunoassay for quantitation of CEA. In fact, a mathematical analysis of the reaction indicates that, under appropriate concentrations of the reactants, circular complexes can be formed which contain the two hybrid antibodies, the GZ enzyme and the CEA antigen. The stability of these complexes can be expected to be substantially greater than that of the more labile CEA-free GZ-antibody complexes, prompting a significant increase in the amount of enzyme molecules which are bound to antibody and are consequently protected from thermal denaturation. These expectations were supported by experimental results: under appropriate conditions, heat-resistant enzyme activity was indeed proportional to concentration of CEA in the range up to 75 ng/ml. As predicted by theory, however, in the presence of excess CEA - in fact at CEA concentrations which are higher than those of possible clinical relevance - circular complexes tended to open up, leading to a marked prozone effect.

Mathematical modelling of tumour growth and treatment

We review some of the models that have been proposed to describe tumour growth and treatment. A first class is that of models which include the analysis of stresses. Here the question of blood vessel collapse in vascular tumours is treated briefly. Results on the existence of radially- and of non-radially-symmetric solutions are illustrated together with an investigation of their stability. Two sections are devoted to tumour cords (growing directly around a blood vessel), highlighting basic facts that are indeed important in the evolution of solid tumours in the presence of necrotic regions. Tumour cords are also taken as an example to deal with certain aspects of tumour treatment. The latter subject is too large to be treated exhaustively but a brief account of the mathematical modelling of hyperthermia is given.